Superelements and Global-Local Analysis

10 Superelements and Gloal-Local Analysis 10 1

Chapter 10: SUPERELEMENTS AND GLOBAL-LOCAL ANALYSIS TABLE OF CONTENTS Page 10.1 Superelement Concept................. 10 3 10.1.1 Where Does the Idea Come From?.......... 10 3 10.1.2 Sudomains.................. 10 5 10.1.3 *Mathematical Requirements........... 10 5 10.2 Static Condensation.................. 10 5 10.2.1 Condensation y Explicit Matrix Operations...... 10 5 10.2.2 Condensation y Symmetric Gauss Elimination...... 10 6 10.2.3 Recovery of Internal Freedoms........... 10 8 10.3 Gloal-Local Analysis.................. 10 8 10. Notes and Biliography................. 10 10 10. References...................... 10 10 10. Exercises...................... 10 11 10 2

10.1 SUPERELEMENT CONCEPT 10.1. Superelement Concept A superelement is a grouping of finite elements that, upon assemly, may e regarded as an individual element for computational purposes. These may e driven y modeling or processing needs. A random assortment of elements does not necessarily make up a superelement. To e considered as such, a grouping must meet certain conditions. Informally we can say that it must form a structural component on its own. This imposes certain conditions stated mathematically in 10.1.3. Inasmuch as these conditions involve advanced concepts such as rank sufficiency, which are introduced in later Chapters, the restrictions are not dwelled upon here. As noted in Chapter 6, superelements may originate from two overlapping contexts: ottom up or top down. In a ottom up context one thinks of superelements as uilt from simpler elements. In a top-down context, superelements may e thought as eing large pieces of a complete structure. This dual viewpoint motivates the following classification: Macroelements. These are superelements assemled with a few primitive elements. Also called mesh units when they are presented to program users as individual elements. Sustructures. Complex assemlies of elements that result on reaking up a structure into distinguishale portions. When does a sustructure ecomes a macroelement or vice-versa? There are no precise rules. In fact the generic term superelement was coined to cover the entire spectrum, ranging from individual elements to complete structures. This universality is helped y common processing features. Both macroelements and sustructures are treated exactly the same way as regards matrix processing. The asic rule is that associated with condensation of internal degrees of freedom. The technique is illustrated in the following section with a simple example. The reader should note, however, that condensation applies to any superelement, whether composed of two or a million elements. 1 10.1.1. Where Does the Idea Come From? Sustructuring was invented y aerospace engineers in the early 1960s 2 to carry out a first-level reakdown of complex systems such as a complete airplane, as depicted in Figure 10.1. The decomposition may continue hierarchically through additional levels as illustrated in Figure 10.2. The concept is also natural for space vehicles operating in stages, such as the Apollo short stack depicted in Figure 10.3. Three original motivating factors for sustructuring can e cited. 1. Facilitate division of laor. Sustructures with different functions are done y separate design groups with specialized knowledge and experience. For instance an aircraft company may set up a fuselage group, a wing group, a landing-gear group, etc. These groups are thus protected from hurry up and wait constraints. More specifically: a wing design group can keep on working on refinements, improvements and experimental model verification as long as 1 Of course the computer implementation ecomes totally different as one goes from macroelements to sustructures, ecause efficient processing for large matrix systems requires exploitation of matrix sparsity. 2 See Notes and Biliography at the end of this Chapter. 10 3

Chapter 10: SUPERELEMENTS AND GLOBAL-LOCAL ANALYSIS S2 S4 S6 S5 S1 S3 Figure 10.1. Complete airplane roken down into six level one sustructures identified as S 1 through S 6. level two sustructure level one sustructure individual element Figure 10.2. Further reakdown of wing structure. The decomposition process may continue down to the individual element level. the interface information (the wing-fuselage intersection) stays sensily unchanged. For some highley complex systems, different sustructures might even e done y separate companies. 2. Take advantage of repetition. Often structures are uilt of several identical or nearly identical units. For instance, COMMAND MODULE the wing sustructures S 2 and S 3 of Figure 10.1 are mirror images on reflection aout the fuselage midplane, and so SERVICE MODULE are the stailizers S 4 and S 5. Even if the loading is not symmetric aout that midplane, recognizing repetitions ADAPTER saves model preparation time. 3. Overcome computer limitations. The computers of the 1960s operated under serious memory limitations. For example, the first supercomputer: the Control Data 6600, had a total high-speed memory of 131072 60-it words or 1.31 MB; that machine cost $10M in 1966 dollars, which would e roughly $100M today (2017). It was difficult to fit a complex structure such as an airplane as one entity. Sustructuring permitted the complete analysis to e carried out in stages through use of auxiliary storage devices such as tapes or disks. 10 4 LUNAR MODULE INSTRUMENT UNIT THIRD STAGE SIV-B Figure 10.3. The Apollo short stack.

10.2 STATIC CONDENSATION Of the three motivations, the first two still hold today. The third one has moved to a different plane: parallel processing, as noted in 10.1.2 elow. In the late 1960s the idea was picked up and developed extensively y the shipuilding industry, the products of which tend to e modular and repetitive to reduce farication costs. (See Note and Biliography.) As previously noted, repetition favors the use of sustructuring techniques. At the other end of the superelement spectrum, the mesh units herein called macroelements appeared in the mid 1960s. They were motivated y user convenience. For example, in hand preparation of models, quadrilateral and ricks involve less human laor than triangles and tetrahedra, respectively. It was therefore natural to comine the latter to assemle the former. Going a step further one can assemle components such as ox elements for applications such as ox-girder ridges. 10.1.2. Sudomains Applied mathematicians working on solution procedures for parallel computation have developed the concept of sudomains. These are groupings of finite elements that are entirely motivated y computational considerations. They are sudivisions of the finite element model done more or less automatically y a program called domain decomposer. Although the concepts of sustructures and sudomains overlap in many respects, it is etter to keep the two separate. The common underlying theme is divide and conquer ut the motivation is different. 10.1.3. *Mathematical Requirements A superelement is said to e rank-sufficient if its only zero-energy modes are rigid-ody modes. Equivalently, the superelement does not possess spurious kinematic mechanisms. Verification of the rank-sufficient condition guarantees that the static condensation procedure descried elow will work properly. 10.2. Static Condensation Degrees of freedom of a superelement are classified into two groups: Internal Freedoms. Those that are not connected to the freedoms of another superelement. Nodes whose freedoms are internal are called internal nodes. Boundary Freedoms. These are connected to at least another superelement. They usually reside at oundary nodes placed on the periphery of the superelement. See Figure 10.4. The ojective is to get rid of all displacement degrees of freedom associated with internal freedoms. This elimination process is called static condensation, or simply condensation. Condensation may e presented in terms of explicit matrix operations, as shown in the next susection. A more practical technique ased on symmetric Gauss elimination is discussed in 10.2.2. 10.2.1. Condensation y Explicit Matrix Operations To carry out the condensation process, the assemled stiffness equations of the superelement are partitioned as follows: [ ][ ] [ ] K K i u f =. (10.1) K i K u i 10 5 f i

Chapter 10: SUPERELEMENTS AND GLOBAL-LOCAL ANALYSIS (a) () i i Figure 10.4. Classification of superelement freedoms into oundary and internal. (a) shows the vertical stailizer sustructure S 6 of Figure 10.2. (The FE mesh is pictured as two-dimensional for illustrative purposes; for an actual aircraft it will e three dimensional.) Boundary freedoms are those associated to the oundary nodes laeled (shown in red), which are connected to the fuselage sustructure. () shows a quadrilateral macroelement mesh-unit faricated with 4 triangles: it has one interior and four oundary nodes. in which suvectors u and u i collect oundary and interior degrees of freedom, respectively. Take the second matrix equation: K i u + K u i = f i, (10.2) If K is nonsingular we can solve for the interior freedoms: u i = K 1 (f i K i u ), (10.3) Replacing into the first matrix equation of (10.1) yields the condensed stiffness equations K u = f. (10.4) In this equation, K = K K i K 1 K i, f = f K i K 1 f i, (10.5) are called the condensed stiffness matrix and force vector, respectively, of the sustructure. From this point onward, the condensed superelement may e viewed, from the standpoint of further operations, as an individual element whose element stiffness matrix and nodal force vector are K and f, respectively. Often each superelement has its own local coordinate system. A transformation of (10.5) to an overall gloal coordinate system is necessary upon condensation. In the case of multiple levels, the transformation is done with respect to the next-level superelement coordinate system. This coordinate transformation procedure automates the processing of repeated portions. Remark 10.1. The feaslity of the condensation process (10.3) (10.5) rests on the non-singularity of K. This matrix is nonsingular if the superelement is rank-sufficient in the sense stated in 10.1.3, and if fixing the oundary freedoms precludes all rigid ody motions. If the former condition is verified ut not the latter, the superelement is called floating. Processing floating superelements demands more advanced computational techniques, among which one may cite the use of projectors and generalized inverses [258]. 10 6

10.2 STATIC CONDENSATION CondenseLastFreedom[K_,f_]:=Module[{pivot,c,Kc,fc, n=length[k]}, If [n<=1,return[{k,f}]]; Kc=Tale[0,{n-1},{n-1}]; fc=tale[0,{n-1}]; pivot=k[[n,n]]; If [pivot==0, Print["CondenseLastFreedom:", " Singular Matrix"]; Return[{K,f}]]; For [i=1,i<=n-1,i++, c=k[[i,n]]/pivot; fc[[i]]=f[[i]]-c*f[[n]]; For [j=1,j<=i,j++, Kc[[j,i]]=Kc[[i,j]]=K[[i,j]]-c*K[[n,j]] ]; ]; Return[Simplify[{Kc,fc}]] ]; K={{6,-2,-1,-3},{ -2,5,-2,-1},{ -1,-2,7,-4},{-3,-1,-4,8}}; f={3,6,4,0}; Print["Before condensation:"," K=",K//MatrixForm," f=",f//matrixform]; {K,f}=CondenseLastFreedom[K,f];Print["Upon condensing DOF 4:", " K=",K//MatrixForm," f=",f//matrixform]; {K,f}=CondenseLastFreedom[K,f];Print["Upon condensing DOF 3:", " K=",K//MatrixForm," f=",f//matrixform]; Figure 10.5. Mathematica module to condense the last degree of freedom from a stiffness matrix and force vector. The test statements carry out the example (10.6) (10.10). 10.2.2. Condensation y Symmetric Gauss Elimination In the computer implementation of the the static condensation process, calculations are not carried out as outlined aove. There are two major differences. The equations of the sustructure are not actually rearranged, and the explicit calculation of the inverse of K is avoided. The procedure may e in fact coded as a variant of symmetric Gauss elimination. To convey the flavor of this technique, consider the following stiffness equations of a superelement: 6 2 1 3 u 1 3 2 5 2 1 u 2 6 =. (10.6) 1 2 7 4 u 3 4 3 1 4 8 u 4 0 Suppose that the last two displacement freedoms: u 3 and u 4, are classified as interior and are to e statically condensed out. To eliminate u 4, perform symmetric Gauss elimination of the fourth row and column: or 6 ( 3) ( 3) 2 ( 1) ( 3) 1 ( 4) ( 3) 8 8 8 2 ( 3) ( 1) 5 ( 1) ( 1) 2 ( 4) ( 1) 8 8 8 1 ( 3) ( 4) 2 ( 1) ( 4) 7 ( 4) ( 4) 8 8 8 u 1 u 2 u 3 = 3 0 ( 3) 8 6 0 ( 1) 8 4 0 ( 4) 8, (10.7) 39 19 5 8 8 2 u 1 3 19 39 5 u 8 8 2 2 = 6. (10.8) 5 5 5 u 2 2 3 4 Repeat the process for the third row and column to eliminate u 3 : [ 39 8 ( 5/2) ( 5/2) 19 5 8 ( 5/2) ( 5/2) ][ ] 5 u1 = 19 8 ( 5/2) ( 5/2) 5 39 8 ( 5/2) ( 5/2) 5 10 7 u 2 [ 3 4 ( 5/2) ] 5 6 4 ( 5/2), (10.9) 5

Chapter 10: SUPERELEMENTS AND GLOBAL-LOCAL ANALYSIS or [ 29 8 29 8 29 8 29 8 ][ u1 u 2 ] = [ ] 5. (10.10) 8 These are the condensed stiffness equations. Figure 10.5 shows a Mathematica program that carries out the foregoing steps. Module CondenseLastFreedom condenses the last freedom of a stiffness matrix K and a force vector f. It is invoked as { Kc,fc }=CondenseLastFreedom[K,f]. It returns the condensed stiffness Kc and force vector fc as new arrays. To do the example (10.6) (10.10), the module is called twice, as illustrated in the test statements of Figure 10.5. Oviously this procedure is much simpler than going through the explicit matrix inverse. Another important advantage of Gauss elimination is that equation rearrangement is not required even if the condensed degrees of freedom do not appear sequentially. For example, suppose that the assemled superelement contains originally eight degrees of freedom and that the freedoms to e condensed out are numered 1, 4, 5, 6 and 8. Then Gauss elimination is carried out over those equations only, and the condensed (3 3) stiffness and (3 1) force vector extracted from rows and columns 2, 3 and 7. An implementation of this process is considered in Exercise 10.2. Remark 10.2. The symmetric Gauss elimination procedure, as illustrated in steps (10.6) (10.10), is primarily useful for macroelements and mesh units, since the numer of stiffness equations for those typically does not exceed a few hundreds. This permits the use of full matrix storage. For sustructures containing thousands or millions of degrees of freedom such as in the airplane example the elimination is carried out using more sophisticated sparse matrix algorithms; for example that descried in [227]. Remark 10.3. The static condensation process is a matrix operation called partial inversion or partial elimination that appears in many disciplines. Here is the general form. Suppose the linear system Ax = y, where A is n n square and x and y are n-vectors, is partitioned as [ ][ ] A11 A 12 x1 = A 22 x 2 A 21 [ y1 ]. (10.11) y 2 Assuming the appropriate inverses to exist, then the following are easily verified matrix identities: [ A 1 11 A 1 11 A ][ ] [ ] [ 12 y1 x1 A11 A 12 A 1 22 A 21 A 1 11 A 22 A 21 A 1 11 A =, A ][ ] [ ] 21 A 12 A 1 22 x1 y1 12 x 2 y 2 A 1 22 A 21 A 1 =. 22 y 2 x 2 (10.12) We say that x 1 has een eliminated or condensed out in the left identity and x 2 in the right one. In FEM applications, it is conventional to condense out the ottom vector x 2, so the right identity is relevant. If A is symmetric, to retain symmetry in (10.12) it is necessary to change the sign of one of the suvectors. 10.2.3. Recovery of Internal Freedoms Once superelements are assemled, a linear solver is called to compute the overall solution vector. As may e expected, this vector contains only oundary freedoms. The recovery of internal freedoms for each superelement is done via (10.3), which means that matrices K and K i must e saved. This process may e carried out either directly through a sequence of matrix operations, or equation y equation as a acksustitution process. 10 8

10.3 GLOBAL-LOCAL ANALYSIS coarse mesh finer meshes Figure 10.6. Left: example panel structure for gloal-local analysis. Right: a FEM mesh for a one-shot analysis. 10.3. Gloal-Local Analysis As discussed in the first Chapter, complex engineering systems are often modeled in a multilevel fashion following the divide and conquer approach. The superelement technique is a practical realization of that approach. A related, ut not identical, technique is multiscale analysis. The whole system is first analyzed as a gloal entity, discarding or passing over details deemed not to affect its overall ehavior. Local details are then analyzed using the results of the gloal analysis as oundary conditions. The process can e continued into the analysis of further details of local models. And so on. When this procedure is restricted to two stages and applied in the context of finite element analysis, it is called gloal-local analysis in the FEM literature. In the gloal stage the ehavior of the entire structure is simulated with a finite element model that necessarily ignores details such as cutouts or joints. These details do not affect the overall ehavior of the structure, ut may have a earing on safety. Such details are a posteriori incorporated in a series of local analyses. The gist of the gloal-local approach is explained in the example illustrated in Figures 10.6 and 10.7. Although the structure is admittedly too simple to merit the application of gloal-local analysis, it serves to illustrate the asic ideas. Suppose one is faced with the analysis of the rectangular panel shown on the top of Figure 10.6, which contains three small holes. The ottom of that figure shows a standard (one-stage) FEM treatment using a largely regular mesh that is refined near the holes. Connecting the coarse and fine meshes usually involves using multifreedom constraints ecause the nodes at mesh oundaries do not match, as depicted in that figure. Figure 10.7 illustrates the gloal-local analysis procedure. The gloal analysis is done with a coarse ut regular FEM mesh that ignores the effect of the holes. This is followed y local analysis of the region near the holes using refined finite element meshes. The key ingredient for the local analyses is the application of oundary conditions (BCs) on the finer mesh oundaries. These BCs may e of displacement (essential) or of force (natural) type. If the former, the applied oundary displacements are interpolated from the gloal mesh solution. If the latter, the internal forces or stresses otained from the gloal calculation are converted to nodal forces on the fine meshes 10 9

Chapter 10: SUPERELEMENTS AND GLOBAL-LOCAL ANALYSIS Gloal analysis with a coarse mesh, ignoring holes, followed ylocal analysis of the vicinity of the holes with finer meshes: Figure 10.7. Gloal-local analysis of prolem of Figure 10.6. through a lumping process. The BC choice noted aove gives rise to two asic variations of the gloal-local approach. Experience accumulated over several decades 3 has shown that the stress-bc approach generally gives more reliale answers. The gloal-local technique can e extended to more than two levels, in which case it receives the more encompassing name multiscale analysis. Although this generalization is still largely in the realm of research, it is receiving increasing attention from various science and engineering communities for complex products such as the thermomechanical analysis of microelectronic components. Notes and Biliography Sustructuring was invented y aerospace engineers in the early 1960s. Przemieniecki s ook [665] contains a fairly complete iliography of early work. Most of this was in the form of largely inaccessile internal company or la reports and so the actual history is difficult to trace. Macroelements appeared simultaneously in many of the early FEM codes. Quadrilateral macroelements faricated with triangles are descried in [221]. For a survey of uses of the static condensation algorithm in FEM, see [882]. The generic term superelement was coined in the late 1960s y the SESAM group at DNV Veritas [209]. The matrix form of static condensation for a complete structure is presented in [26], as a scheme to eliminate unloaded DOF in the displacement method. It is unclear when this idea was first applied to sustructures or macroelements. The first application for reduced dynamical models is y Guyan [356]. The application of domain decomposition to parallel FEM solvers has produced an enormous and highly specialized literature. Procedures for handling floating superelements using generalized inverse methods are discussed in [250,258,630]. The gloal-local analysis procedure descried here is also primarily used in industry and as such it is rarely mentioned in academic textooks. References Referenced items have een moved to Appendix R. 3 Particularly in the aerospace industry, in which the gloal-local technique has een used since the early 1960s. 10 10

Exercises Homework Exercises for Chapter 10 Superelements and Gloal-Local Analysis EXERCISE 10.1 [N:15] The free-free stiffness equations of a superelement are 88 44 44 0 u 1 5 44 132 44 44 u 44 44 176 44 2 10 u = 3 15. 0 44 44 220 u 4 20 (E10.1) Eliminate u 2 and u 3 from (E10.1) y static condensation, and show (ut do not solve) the condensed equation system. Use either the explicit matrix inverse formulas 10.5 or the symmetric Gauss elimination process explained in 10.2.2. Hint: regardless of method the result should e [ ][ ] [ ] 52 36 u1 15 =. (E10.2) 36 184 u 4 30 EXERCISE 10.2 [C+N:20] Generalize the program of Figure 10.5 to a module CondenseFreedom[K,f,k] that is ale to condense the kth degree of freedom from K and f, which is not necessarily the last one. That is, k may range from 1 to n, where n is the numer of freedoms in Ku = f. Apply that program to solve the previous Exercise y calling it twice, first with k=2 and then with k=2. (Why 2 oth times?) Hint: here is a possile way of organizing the inner loop in Mathematica (after checking that the pivot is nonzero): =0; For [i=1,i<=n,i++, If [i==k, Continue[]]; ++; c=k[[i,k]]/pivot; fc[[]]=f[[i]]-c*f[[k]]; jj=0; For [j=1,j<=n,j++, If [j==k, Continue[]]; jj++; Kc[[,jj]]=K[[i,j]]-c*K[[k,j]] ]; ]; Return[{Kc,fc}] An implementation of this fragment in pseudo-code is shown in Figure E10.1. =0; for i=1,i<=n,i++, if i==k, continue; endif; ++; c=k(i,k)/pivot; fc()=f(i)-c*f(k); jj=0; for j=1,j<=n,j++, if j==k, continue; endif; jj++; Kc(,jj)=K(,jj) c*k(k,j); endfor; // loop over columns endfor; // loop over rows return(kc,fc); Figure E10.1. Pseudo-code for condensing an aritrary freedom (no necessarily the last one). EXERCISE 10.3 [D:15] Explain the similarities and differences etween superelement analysis and gloallocal FEM analysis. 10 11

Chapter 10: SUPERELEMENTS AND GLOBAL-LOCAL ANALYSIS EXERCISE 10.4 [A:20] If the superelement stiffness K is symmetric, the static condensation process can e viewed as a special case of the master-slave transformation method discussed in Chapter 8. To prove this, take exterior freedoms u as masters and interior freedoms u i as slaves. Assume the master-slave transformation relation u = [ u ] [ = u i ] I K 1 K i [ [ u ] 0 K 1 f i ] def = Tu g. (E10.3) Work out ˆK = T T KT and ˆf = T T (f Kg), and show that ˆKû = ˆf coalesces with the condensed stiffness equations (10.4) (10.5) if u û. (Take advantage of the symmetry properties K T i = K i, (K 1 ) T = K 1.) EXERCISE 10.5 [D:30] (Requires thinking) Explain the conceptual and operational differences etween one-stage FEM analysis and gloal-local analysis of a prolem such as that illustrated in Figures 10.6 and 10.7. Are the answers the same? What is gained, if any, y the gloal-local approach over the one-stage FEM analysis? EXERCISE 10.6 [N:20] The widely used Guyan s scheme [356] for dynamic model reduction applies the static-condensation relation (E10.3) as master-slave transformation to oth the stiffness and the mass matrix of the superelement. Use this procedure to eliminate the second and third DOF of the mass-stiffness system: 2 1 0 0 1 1 0 0 1 4 1 0 1 2 1 0 M = 0 1 4 1, K = 0 1 2 1. (E10.4) 0 0 1 2 0 0 1 1 Compute and compare the viration frequencies of the eigensystems (ωi 2M K)v i = 0 and ( ˆω i 2 ˆM ˆK)ˆv i = 0 efore and after reduction, respectively. Hints. The four original squared frequencies are the eigenvalues of M 1 K, which may e otained, for example, with the Matla eig function. The largest is 2, the lowest 0. To perform the Guyan reduction it is convenient to reorder M and K so that rows and columns 2 and 3 ecome 3 and 4, respectively: 2 0 1 0 v 1 1 0 1 0 v 1 ω 2 0 2 0 1 v 4 0 1 0 1 v 1 0 4 1 = 4 1 0 2 1, (E10.5) 0 1 1 4 0 1 1 2 v 2 v 3 v 2 v 3 which in partitioned matrix form is ω 2 [ M M i ][ ] [ ][ ] M i v K K = i v. (E10.6) M v i K v i K i Next show that static condensation of K is equivalent to a master-slave transformation 1 0 v [ ] 1 1 0 [ ] I 0 1 v T = K 1 = K relating 4 0 1 v1 =. (E10.7) i v 4 2/3 1/3 1/3 2/3 v 2 v 3 2/3 1/3 1/3 2/3 The rest is easy. Form ˆM = T T MT and ˆK = T T KT, find the two squared frequencies of the condensed eigensystem and verify that they approximate the two lowest original squared frequencies. 10 12

Exercises (a) F () F 1 1 y x 2 2k 2k 2k 45 45 45 45 2k 2k 3 reduced model y static condensation k eq 4 4 Figure E10.2. Model reduction of spring assemly y static condensation, for Exercise 10.7: (a) original five-spring assemly with five DOF; () equivalent spring model with only one DOF. EXERCISE 10.7 [C:20] Consider the spring assemly shown in Figure E10.2(a). It has 4 nodes connected y 5 spring elements, all of which have axial stiffness 2k. 4 Upon explicit removal of the 3 support-constrained equations, the system has 5 degrees of freedom: u y1, u x2, u y2, u x3, and u y3. The master stiffness equation for the retained freedoms is 2 1 1 1 1 1 4 0 2 0 k 1 0 2 0 0 1 2 0 4 0 1 0 0 0 2 u y1 u x2 u y2 u x3 u y3 = F 0 0 0 0. (E10.8) The expression (E10.8) is considered given. If one assumes that nodes 2 and 3 are not connected to other elements, their four DOF: u x2 through u y3, can e eliminated y static condensation to produce the reduced model pictured in Figure E10.2(). The stiffness equation of this model is simply k eq u y1 = F, (E10.9) in which k eq is called an equivalent spring. Show that k eq = 2k/3. 5 EXERCISE 10.8 [C:25] Solve the foregoing prolem for an aritrary angle 0 <φ<90 in lieu of 45. (Symolic computation is now mandatory.) EXERCISE 10.9 [A:20] Two eam elements: 1 2 and 2 3, each of length L and rigidity EI are connected at node 2. The macroelement has 6 degrees of freedom: {v 1,θ 1,v 2,θ 2,v 3,θ 3 }. Eliminate the two DOF of node 2 y condensation. Is the condensed stiffness the same as that of a eam element of length 2L and rigidity EI? (For expressions of the eam stiffness matrices, see Chapter 12.) 4 The factor of 2 avoids fractions in the coefficient matrix of (E10.8). 5 The spring lengths are irrelevant to the result; only the angle here 45 is important. 10 13